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Asymptotic wave solutions
Jean Virieux
Professeur UJF
Translucid Earth Source
Receiver
Same shape !
T(x)
Too diffracting medium : wavefront coherence lost !
Wavefront preserved
Wavefront : T(x)=T0
Travel-time T(x)
Amplitude A(x)
S(t)
( )
( , ) ( ) ( ( ))
( , ) ( ) ( ) i T x
u x t A x S t T x
u x A x S e
Eikonal equation
)(
1)(
)(
1)(
xcxT
xcL
T
T
Lxc x
Two simple interpretation of wavefront evolution
Orthogonal trajectories are rays
T+DT
T=cte
Velocity c(x)
DL
Grad(T) orthogonal to wavefront
Direction ? : abs or square )(
1))((
22
xcxTx
Transport Equation
0).2(
0)()(0
...0
..0
..
2
22
211
'1
21222
22
1112
12222
2
2222
21112
1
222
2112
1
TATAA
TAdivdTAdiv
dSnTAdSnTAdSnTA
dSnTAdSnTA
dSnTAdSnTA
TdSATdSAd
cccc
Tracing neighboring rays defines a ray tube : variation of amplitude depends on section variation
0)()()().(2 2 xTxAxTxA
Ray tracing equations
1 tds
xdt
ds
xd
)(sx
ds
Td
)1
(2
)(2 2
2
c
cT
c
ds
Td
))(
1()
)(
1(
xcds
xd
xcds
d
Ray
T
T=cte
s curvilinear abscisse
)()(// xTxcds
xdT
ds
xd
evolution of x
Evolution of is given by butT
.)(. Txctds
d )(. TTcds
Td
therefore
)1
(cds
Td
evolution of T
Curvature equation known as the ray equation
)(xTp
)(sxq
We often note the slowness vector and the position
Evolution of
System of ray equations
1
dqcp
dsdp
ds c
1 1
dqp
d
dp
d c c
2
1
dqc p
dTdp
cdT c
ddTc
dscdT
cdsd
2
with
which ODE to select for numerical solving ? Either T or x sampling
Many analytical solutions (gradient of velocity; gradient of slowness square)
Velocity variation v(z)
dz
zduzu
d
dp
d
dp
d
dp
pd
dqp
d
dqp
d
dq
zyx
zz
yy
xx
)()(;0;0
;;
Ray equations are
The horizontal component of the slowness vector is constant: the trajectory is inside a plan which is called the plan of propagation. We may define the frame (xoz) as this plane.
dz
zduzu
d
dp
d
dp
pd
dqp
d
dq
zx
zz
xx
)()(;0
;
22 )( x
x
z
x
z
x
pzu
p
p
p
dq
dq
Where px is a constante
1
0
220011
)(),(),(
z
z x
xxxxx dz
pzu
ppzqpzq
For a ray towards the depth
Velocity variation v(z)
)(222px zupp
pp
pp
z
z x
z
z x
x
z
z x
x
z
z x
xxxx
dzpzu
zudz
pzu
zuTpzT
dzpzu
pdz
pzu
pqpzq
10
10
22
2
22
2
011
2222011
)(
)(
)(
)(),(
)()(),(
p
p
z
z
dzpzu
zupT
dzpzu
ppX
022
2
022
)(
)(2)(
)(2)(
At a given maximum depth zp, the slowness vector is horizontal following the equation
zp
If we consider a source at the free surface as well as the receiver, we get
2 2 2
2 2
2 2 2
2( )
( )2
( )
sin
p
p
a
r
a
r
p dr
rr u r p
r u r drT
rr u r p
with p ru i
In Cartesian frame In Spherical frame
with p = usini
Velocity structure in the Earth
Radial Structure
Main discontinuous boundaries Crustal discontinuity (Mohorovicic (moho – 30 km) Mantle discontinuity (Gutenberg – 2900 km) D’’ Core discontinuity (Lehman – 5100 km)
Shadow zone at the Gutenberg discontinuity: thickness of few kms
Minor discontinuities
Interface at 100-200 km Interface at 670-700 km Interface at 15 km (discontinuity of Conrad)
These discontinuities are related to lithospheric, mesospheric and sismogenic structures.
They are not expected to be deployed over the entire globe
System of ray equations
cds
pd
pcds
xd
1
ccd
pd
pd
xd
11
cc
dT
pd
pcdT
xd
1
2
ddTc
dscdT
cdsd
2
with
which ODE to select for numerical solving ? Either T or x sampling
Many analytical solutions (gradient of velocity; gradient of slowness square)
( )dy
A yd
Non-linear ODE !
Time integration of ray equations
Initial conditions EASY
1D sampling of 2D/3D medium : FAST
source
receiver
Runge-Kutta second-order integration
Predictor-Corrector integration stiffness
source
receiver Boundary conditions VERY DIFFICULT
?
?
Shooting dp ?
Bending dx ?
Continuing dc ?
AND FROM TIME TO TIME IT FAILS !
But we need 2 points ray tracing because we have a source and a receiver to be connected !
Save p conditions if possible !
Eikonal Solvers Fast marching method (FMM)
22
222
)(1
1)()(
x
T
cz
T
cz
T
x
T
z
T
x
T
(tracking interface/wavefront motion : curve and surface evolution)
Layered medium
Let us assume T is known at a level z=cte
z=cte
z
z + dz
Compute along z=cte by a finite difference
approximationz
T
Deduce and extend T estimation at depth z+dz
x
T
Assuming 1D plane wave propagation, we have been able to estimate T at a depth z+dz
From Sethian, 1999
EIKONAL SOLVER
Fast marching method (FMM) 2 2
2
22
1( ) ( )
( , )
1( )
( , )
T T
x z c x z
T T
z c x z x
(tracking interface/wavefront motion : curve and surface evolution)
2D case
Strategy available in 2D and 3D BUT only for the minimum time in each node in the spatial domain (x,y,z).
Possible extension in the phase domain (for multiphases) ?
Sharp interfaces are always difficult to handle in this discrete formulation
From Sethian, 1999
Other techniques as fast sweeping methods3D case
Wavefront trackingBack-raytracing strategy
Once traveltime T is computed over the grid for one source, we may backtrace using the gradient of T from any point of the medium towards the source (should be applied from each receiver)
The surface {MIN TIME} is convex as time increases from the source : one solution !
A VERY GOOD TOOL FOR First Arrival Time
Tomography (FATT)
Time over the grid Ray
Smooth medium : simple case
Rays and wavefronts in an homogeneous medium. (Lambaré et al., 1996)
Ray tracing by wavefronts
Rays and wavefronts in a constant gradient of the velocity. (Lambaré et al., 1996)
(Lambaré et al., 1996)
Rays and wave fronts in a complex medium. (Lambaré et al., 1996)
(Lambaré et al., 1996)
(Lambaré et al., 1996)
(Lambaré et al., 1996)
0 400 800 1200 1600 2000
Z in m
Ray tracing by wavefronts
Evolution over time :
folding of the wavefront is allowed
Dynamic sampling :
undersampling of ray fans
oversampling of ray fans
Keep sampling of the medium by rays « uniform »
Heavy task 2D & 3D !
Example of wavefront evolution in Marmousi model
Hamilton Formulation
pp
0
0 q
p
Information around the ray
Ray
2
1
2
1
cd
pd
pd
qd
)1
(2
1),(
22
cppqH
Hd
pd
Hd
qd
x
p
mechanics : ray tracing is a balistic problem
sympletic structure (FUN!)
Meaning of the neighbooring zone – Fresnel zone for example but also anything you wish
Paraxial Ray theory
yyd
d
)( y
Estimation of ray tube : amplitude
Estimation of taking-off angles : shooting strategy…
does not depend on : LINEAR PROBLEM (SIMPLE) !
qpqHppqHd
qd
ppqqHd
qqd
qppp
pp
),(),(
),()(
0000
000
0000
0
0 0
0 0pq pp
qq qp
H Hq qdH Hp pd
Practical issues
2 2, ,2 2, ,
0 0 1 0
0 0 0 1
0.5 0.5 0 0
0.5 0.5 0 0
x x
z z
xx xzx x
zx zzz z
q q
q qdu up pd
u up p
Four elementary paraxial trajectories
dy1t(0)=(1,0,0,0)
dy2t(0)=(0,1,0,0)
dy3t(0)=(0,0,1,0)
dy4t(0)=(0,0,0,1)
, , ,tx z x zy q q p p
NOT A RAY !
Practical issues 0H
From paraxial trajectories, one can combine them for paraxial rays as long as the perturbation of the Hamiltonian is zero.
For a point source, a small slowness perturbation (a=10-4) gives a good approximate ray depending on the velocity variation in the medium: it is a kind of derivative ….
(0) (0)
(0) (0)x z
z x
p p
p p
Paraxial rays require other conservative quantities : the perturbation of the Hamiltonian should be zero (or, in other words, the eikonal perturbation is zero)
0x x z zp p p p
2 2, ,
1 1( , ) ( , ) 0
2 2x x z z x x z zp p p p u x z q u x z q
Point source conditions dqx(0)=dqz(0)=0
( ) (0) 3( ) (0) 4( )z xy p y p y
This is enough to verify this condition initially
Solution
Practical issues
Two independent paraxial rays exist in 2D (while we have four elementary paraxial trajectories)
Point paraxial ray
Plane paraxial ray
2,
2,
(0) ( , ) (0)
(0) ( , ) (0)
x x
z z
q u x z
q u x z
Plane wave conditions dpx(0)=dpz(0)=0
2 2, ,( ) ( , ) (0) 1( ) ( , ) (0) 2( )x zy u x z y u x z y
This is enough to verify this condition initially
Solution
2 2, ,( , ) ( , ) 0x x z zu x z q u x z q
Seismic attributes
Travel time evolution with the grid step : blue for FMM and black for recomputed time
One ray Log scale in time
Grid step
S
R
A ray
2 PT ray tracing non-linear problem solved, any attribute could be computed along this line :
- Time (for tomography)
- Amplitude (through paraxial ODE integration fast)
- Polarisation, anisotropy and so on
Moreover, we may bend the ray for a more accurate ray tracing less dependent of the grid step (FMM)
Keep values of p at source and receiver !
Time error over the grid (0)
NOT THE SAME COLOR SCALE (factor 100)
Coarser grid for computation
Errors through FMM times Errors through rays deduced after FMM times
ODE resolution Runge-Kutta of second order Write a computer program for an
analytical law for the velocity: take a gradient with a component along x and a component along z
Home work : redo the same thing with a Runge-Kutta of fourth order (look after its definition)
Consider a gradient of the square of slowness
Runge-Kutta integration
1/ 2 0 0 0
1 0 1/ 2 1/ 2
( )
. ( )2
. ( )
dfA f
d
f f A f
f f A f
Second-order integration
Non-linear ray tracing
Second-order euler integration for paraxial ray tracing
1 0 0
.
. .
dfA f
d
f f A f
2 2, ,2 2, ,
0 0 1 0
0 0 0 1
0.5 0.5 0 0
0.5 0.5 0 0
x x
z z
xx xzx x
zx zzz z
q q
q qdu up pd
u up p
Four elementary paraxial trajectories
dy1t(0)=(1,0,0,0)
dy2t(0)=(0,1,0,0)
dy3t(0)=(0,0,1,0)
dy4t(0)=(0,0,0,1)
, , ,tx z x zy q q p p
Remember !
3 is for point solution along x
4 is for point solution along z
Two points ray tracing: the paraxial shooting method
. where the shooting angle is .xi
i
d qx
d
Consider Dx the distance between ray point at the free surface and sensor position
0 0 0
0 0 0
3 4
0
x zx x i x i
x zi i i i iz z z
z xx x xi ix z
i i iz z z
z xxi i
i z
d q d q dp d q dp
d dp d dp d
d q d q d qp p
d dp dp
d qqx p qx p
d
We can compute an estimated Dq and, therefore, a new shooting angle
The estimation of the derivative is through paraxial computation
Amplitude estimation
2 2
3 4 3 4
2 2
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
x z z x
x z
z x z xx i x i z z i z i x
x z
q p r q p rL
p r p r
q p q p p r q p q p p rL
p r p r
Consider DL the distance between the exit point of a ray and the paraxial ray running point.
DL
Thanks to the point paraxial ray estimation dq3 and dq4, we may estimate the geometrical spreading DL/ Dq and, therefore, the amplitude A(r)
( )L
A r
Stop and move to the numerical exercise
How to reconstruct the velocity structure ?
Forward problem (easy)from a known velocity structure, it is possible to compute
travel times, emergent distance and amplitudes.
Inverse problem (difficult)from travel times (or similarly emergent distances), it is
possible to deduce the velocity structure : this is the time tomography
even more difficult is the diffraction tomography related to the waveform and/or the preserved/true amplitude.
Tomographic approach
Very general problem
medicine; oceanography, climatology
Difficult problem when unknown a priori medium (travel time tomography)
Easier problem if a first medium could be constructed: perturbation techniques can be used for improving the reconstruction (delayed travel time tomography)
Travel Time tomography
We must « invert » the travel time or the emergent distance for getting z(u): we select the distance.
Abel problem (1826)
2
22
2
022
2
20
/
2
)(;
)(2)( du
pu
dudz
p
pXdz
pzu
ppX
p
u
z p
Determination of the shape of a hill from travel times of a ball launched at the bottom of the hill with various initial velocity and coming back at the initial position.
ABEL PROBLEM
A point of mass ONE and initial velocity v0 reaches a maximal height x given by 2
02
1vgx
We shall take as the zero value for the potential energy: this gives us the following equations and its integration:
2
0
/2 ( ) ; ( )
2 ( )
xds ds dg x t x d
dt g x
We may transform it into the so-called Abel integral where t(x) is known and f(x) is the shape of the hill to be found: this is an integrale equation.
d
x
fxt
x
0
)()(
dsdx
x
y
x
The exact inverse solution
0
0
00
00
000
)(1)(
)()(
)()(
)()(
)()(
x
dxxt
d
df
fx
dxxt
d
d
dfdxx
xt
xx
dxdfdx
x
xt
dx
f
x
dxdx
x
xt xWe multiply and we integrate
We inverse the order of integration
We change variable of integration
We differentiate and write it down the final expression
22 sincos x
THE ABEL SOLUTION
a
x
a
x
dxxt
d
df
dx
fxt
)(1)(
)()(
By changing variable x in a-x and x in a-x, we get the standard formulae
We must havet(x) should be continuous,
t(0)=0 t(x) should have a finit derivative with a finite number of discontinuities.The most restrictive assumption is the continuity of the function t(x).
The solution HWB : HERGLOTZ-WIECHERT-BATEMAN
)(
0
22
2
22
2
22
2
)cosh(
1)(
)(1)(
2/)(1)(
/
2
)(
0
2
20
2
20
uX
u
u
u
u
p
u
pv
dXvz
dpup
pXvz
dpup
ppXvz
dupu
dudz
p
pX
In Cartesian frame In Spherical frame
)/(
0 )/cosh(
1)
)(ln(
vr
rpv
d
vr
R
From the direct solution, we can deduce the inverson solutionAfter few manipulations, we can move from the Cartesian expression towards the Spherical expression
We find r(v) as a value of r/v
Stratified medium We may find the interface at a depth h
when considering all waves We may reconstruct an infinity of
structures with only direct and refracted waves.
We have a velocity jump when a velocity decrease
Velocity structure with depth Velocity profile built without any a priori
An difficulty arises when the velocity decreases.
An initial model through the HWB method
An initial model can be built The exact inverse formulae does not allow to introduce
additional information,
F. Press in 1968 has preferred the exhaustive exploration of possible profiles (5 millions !). The quality of the profile is appreciated using a misfit function as the sum of the square of delayed times as well as total volumic mass and inertial moments well constrained from celestial mechanics ...
Exploration through grid search, Monte Carlo search, simulated annealing, genetic algorithm, tabou method, hant search …
The symmetrical radial EARTH
A simple case : small perturbation
Initiale structure of
velocity Search of small variation
of velocity or slowness Linear approach
Example: Massif Central
GLOBAL Tomography
Velocity variation at a depth of 200 km : good correlation with superficial structures.
Velocity variations at a depth of 1325 km : good correlation with the Geoid.
Courtesy of W. Spakman
Delayed Travel-time tomography
0( , ) ( , , ) ( , , ) ( , , )receiver receiver receiver
source source source
t s r u x y z dl u x y z dl u x y z dl
0 0
0 0
0
0
0
0
0
0
( , ) ( , , ) ( , , )
( , ) ( , ) ( , , )
( , ) ( , , )
receiver receiver
source source
receiver
source
receiver
source
t s r u x y z dl u x y z dl
t s r t s r u x y z dl
t s r u x y z dl
Consider small perturbations du(x) of the slowness field u(x)
station
source
dlzyxustationsourcet ),,(),(Finding the slowness u(x) from t(s,r) is a difficult problem: only techniques for one variable !
This a LINEAR PROBLEM dt(s,r)=G(du)
DESCRIPTION OF THE VELOCITY PERTURBATION
The velocity perturbation field (or the slowness field) du(x,y,z) can be described into a meshed cube regularly spaced in the three directions.
For each node, we specify a value ui,j,k. The interpolation will be performed with functions as step funcitons. We have found shape functions h,i,j,k=1 pour i,j,k, and zero for other indices.
cube
kjikji huzyxu ,,,,),,(
Discrete Model Space
cube
kjikji huzyxu ,,,,),,(
m
m
m
nn
m
n
n
cubekji
cube rayon
kjikji
rayon cubekjikji
u
u
u
u
u
t
u
t
u
t
u
t
t
t
t
t
uu
trst
dlhudlhurst
1
2
1
1
1
1
1
1
2
1
,,
,,,,,,,,
...
),(
),(00
Slowness perturbation description
0t G u
Matrice of sensitivity or of partial derivatives
Discretisation of the medium fats the ray
Sensitivity matrice is a sparse matrice
LINEAR INVERSE PROBLEM
1 10 0
0 0
u G t m G d
t G u d G m
Updating slowness perturbation values from time residuals
Formally one can write
with the forward problem
Existence, Uniqueness, Stability, RobustnessDiscretisation
Identifiability
of the model
Small errors propagates
Outliers effects
NON-UNIQUENESS & NON-STABILITY : ILL-POSED PROBLEM
REGULARISATION : ILL-POSED -> WELL-POSED
LEAST-SQUARES SOLUTIONS
AtDT=AtA DM
• The linear system can be recast into a least-square system, which means a system of normal equations. The resolution of this system gives the solution. DM=(AtA)-1AtDT
• The system is both under-determined and over-determined depending on the considered zone (and tne number of rays going through.
LEAST SQUARES METHOD
dGGGm
dGmGG
m
mE
mGdmGdmE
ttest
tt
t
0
1
00
000
00
0)(
)()()(
L2 norm
locates the minimum of E
normal equations
if exists 1
00
GG t
Least-squares estimation
Operator on data will derive a new model : this is called
the generalized inverse
tt GGG 0
1
00
gG0
G0 is a N by M matrice
is a M by M matrice 1
00
GG t
Under-determination M > N
Over-determination N > MMixed-determination – seismic tomography
Maximum Likelihood method
One assume a gaussian distribution of data
Joint distribution could be written
)()(
2
1exp)( 0
10 mGdCmGddp d
Where G0m is the data mean and Cd is the data covariance matrice: this method is very similar to the least squares method
)()()()()()( 01
0100 mGdCmGdmEmGdmGdmE dt
)()()( 01
02 mGdWmGdmE d
Even without knowing the matrice Cd, we may consider data weight Wd through
SVD analysis for stability and uniqueness
SVD decomposition :
U : (N x N) orthogonal Ut=U-1
V : (M x M) orthogonal Vt=V-1
L : (N x M) diagonal matrice Null space for Li=0
tVUG 0
UtU=I and VtV=I (not the inverse !)
][
][
0
0
UUU
VVV
p
p
tpp
p VVUUG 000 00
0
Vp and V0 determine the uniqueness while Up and U0 determine the existence of the solution
tppp
tppp
UVG
VUG11
0
0
Up and Vp have now inverses !
Solution, model & data resolution
RmmVVmVUUVmGGdGm tp
tppp
tpppest 1
01
01
0 )(The solution is
where Model resolution matrice : if V0=0 then R=VVt=I tppVVR
NddUUmGd tppestest 0
dUUN tppwhere Data resolution matrice : if U0=0 then N=UUt=I
importance matriceGoodness of resolution
SPREAD(R)=
SPREAD(N)=
Spreading functions
2
2
IN
IR
Good tools for quality estimation
PRIOR INFORMATION Hard bounds
Prior model
e is the damping parameter controlling the importance of the model mp
Gaussian distribution
Model smoothness
Penalty approach
add additional relations between model parameters (new lines)
)()()()()( 005 pmt
pdt mmWmmmGdWmGdmE
With Wd data weighting and Wm model weighting
tmd
tg
pmt
pdt
GCGCGG
mmCmmmGdCmGdmE
0
110
100
10
104 )()()()()(
)()()()()( 003 pt
pt mmmmmGdmGdmE
BmA i Seismic velocity should be positive
UNCERTAINTY ESTIMATION Least squares solution
Model covariance : uncertainty in the data
curvature of the error function
Sampling the error function around the estimated model often this has to be done numerically
dGdGGGm gttest 00
1
00
1
2
22
1
002
2
0000
2
1cov
cov
covcov
estmm
dest
tdest
dd
gtd
ggtgest
m
Em
GGm
IC
GCGGdGm
Uncorrelated data
A posteriori model covariance matrice True a posteriori distribution
Tangent gaussian distribution
S diagonal matrice eigenvalues
U orthogonal matrice eigenvectors
Error ellipsoidal could be estimated
WARNING : formal estimation related to the gaussian distribution hypothesis
tmd
t USUCGCG 10
10
If one can decompose this matrice
A priori & A posteriori informationWhat is the meaning of the « final » model we provide ?
acceptable
Flow chart
true ray tracing
data residual
sensitivity matrice
model update
new model
mmm
dGm
mgG
ddd
mgd
m
d
synobs
syn
obs
10
0
)(
collected data
starting modelloop
Calculate for formal uncertainty estimation
small model variation or small errors exit
2
2
mE
Sampling a posteriori distribution
Resolution estimation : spike test
Boot-Strapping
Jack-knifing
Natural Neighboring
Monte-Carlo
Sampling a posteriori distribution
Uncertainty estimation for P and S velocities using boot-strapping techniques
Steepest descent methods )()( 1 kk mEmE
kk
kk
kkkk
k
kkk
DmE
mEmEd
dE
Ed
mEd
mEmEtmE
)(
)()(
)(
)())((
2
12
1
0
Gradient method
Conjugate gradient
Newton
Quasi-Newton
Gauss-Newton is Quasi-Newton for L2 norm
quadratic approximation of E
Tomographic descent 2
2/1
2/1
2/1
2/1 )(
2
1
mC
mgC
mC
dC
m
d
pm
dMinimisation of this vector
2/1
2/1
m
kdk
C
GCAIf one computes
then
)(
))((
02/1
2/1
km
kdtkk
tk
mmC
dmgCAmAA
Gaussian error distribution of data and of a posteriori model
Easy implementation once Gk has been computed
Extension using Sech transformation (reducing outliers effects while keeping L2 norm simplicity)
LSQR method
The LSQR method is a conjugate gradient method developped by Paige & Saunders
Good numerical behaviour for ill-conditioned matrices
When compared to an SVD exact solution, LSQR gives main components of the solution while SVD requires the entire set of eigenvectors
Fast convergence and minimal norm solution (zero components in the null space if any)
Corinth Gulf
An extension zone where there is a deep drilling project.
How this rift is opening?
What are the physical mechanisms of extension (fractures, fluides, isostatic equilibrium)
Work of Diana Latorre and of Vadim Monteiller
Seismic experiment 1991 (and one in 2001)
MEDIUM 1D : HWB AND RANDOM SELECTION
Velocity structure imageHorizontal sections
Velocity structure image Vertical sections
P S
Vp/Vs ratio: fluid existence ?
Recovered parameters might have diferent interpretation and the ratio Vp/Vs has a strong relation with the presence of fluids or the relation Vp*Vs may be related to porosity
Other methods of exploration
Grid search Monte-Carlo (ponctual or
continuous) Genetic algorithm Simulated annealing and co Tabou method Natural Neighboring method
Conclusion FATT
Selection of an enough fine grid Selection of the a priori model information Selection of an initial model FMM and BRT for 2PT-RT Time and derivatives estimation LSQR inversion Update the model Uncertainty analysis (Lanzos or numerical)
THANK YOU !
Many figures have come from people I have worked with: many thanks to them !
Kirchhoff approximation
1) Representation theorem
2) Kirchhoff summation
3) Reciprocity
Born approximation
1) Single scattering approximation
2) Surface approximation
(Forgues et al., 1996)
(Forgues et al., 1996)
